Open Phys. 2017; 15:58–67
Research Article
Open Access
F. Alzahrani, Elbaz I. Abouelmagd, Juan L.G. Guirao*, and A. Hobiny
On the libration collinear points in the restricted
three–body problem
DOI 10.1515/phys-2017-0007
Received September 3, 2016; accepted October 3, 2016
Abstract: In the restricted problem of three bodies when
the primaries are triaxial rigid bodies, the necessary and
sufficient conditions to find the locations of the three libration collinear points are stated. In addition, the Linear
stability of these points is studied for the case of the Euler
angles of rotational motion being θ i = 0, ψ i + φ i = π/2,
i = 1, 2 accordingly. We underline that the model studied in this paper has special importance in space dynamics
when the third body moves in gravitational fields of planetary systems and particularly in a Jupiter model or a problem including an irregular asteroid.
Keywords: Restricted three-body problem; triaxial rigid
bodies; libration points; stability
PACS: 02.30.Hq
1 Introduction
In recent years, the importance and significance of the libration points for space applications has increased within
the scientific community. This is because these points are
natural equilibrium solutions of the restricted three-body
problem and offer the unique possibility to obtain a fixed
configuration with respect to two primaries. Thereby, the
solution to libration points could alleviate a lot of mission
constraints which are not realizable with the classical Kep-
F. Alzahrani, A. Hobiny: Nonlinear Analysis and Applied Mathematics Research Group (NAAM)
Department of Mathematics, Faculty of Science, King Abdulaziz
University Jeddah, Saudi Arabia.
Elbaz I. Abouelmagd: Celestial Mechanics Unit, Astronomy Department, National Research Institute of Astronomy and Geophysics
(NRIAG), Helwan 11421, Cairo, Egypt,
E-mail: [email protected] or [email protected]
*Corresponding Author: Juan L.G. Guirao: Departamento de
Matemática Aplicada y Estadística Universidad Politécnica de Cartagena Hospital de Marina 30203 Cartagena, Región de Murcia, Spain.
E-mail: address: [email protected]
lerian two-body orbits. Moreover, exploiting the stable and
unstable part of the dynamics regarding these points, lowenergy interplanetary, moon-to-moon transfers of practical interest can be obtained. Around each of the three
collinear equilibrium points a family of unstable orbits exists, see Abouelmagd et al. [1]. These orbits are useful for
many space applications requiring a fixed configuration
with respect to two primary bodies. The orbits are also useful when the calculation of planar Lyapunov orbits that
emerge from these points is necessary and the ballistically
captured transfers are needed, for more details see Koon
et al. [2]
The model of the three-body problem is used to determine the possible motions of three bodies which attract
each other according to Newton’s law of inverse squares. It
started with Newton’s perturbative studies on the inequalities of the lunar motion. In physics and classical mechanics the problem has two conspicuous meanings:
– In its conventional sense, the problem yields an initial set of data that characterize the positions and
velocities of three bodies at a specified time. In accordance with the laws of classical mechanics, the
motions of the three bodies can be determined.
– In an extended modern sense, the three-body problem is a class of problems in classical or quantum
mechanics that model the motion of three particles.
Historically, the first specific three-body problem receiving extended study was the one involving the lunar
theory, the motion of the Moon under the gravitational influence of the Earth and the Sun. Improving the accuracy
of the lunar theory came to be of topical interest at the end
of the eighteenth century. This interest mainly arose from
the belief that the lunar theory could be applicable to navigation at sea in the development of a method for determining geographical longitude. Following Newton’s work
it was appreciated that a major part of the problem in lunar theory consisted in evaluating the perturbing effect of
the Sun on the motion of the Moon around the Earth.
Some significance of the three-body problem comes
from two links. For the first, it is considered a special case
of the n-body problem, which describes how n objects will
Âľ 2017 F. Alzahrani et al., published by De Gruyter Open.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
Unauthenticated
Download Date | 6/16/17 6:48 PM
On the libration collinear points
move under one of the physical forces, such as gravity.
These problems have an analytical solution in the form of a
convergent power series. For the second link, the problem
can be reduced to a perturbed two body problem. Some applications are conveniently treated by perturbation theory,
in which the system is considered as a two-body problem
plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.
The restricted three-body problem under the effects
of perturbed forces has been receiving considerable attention from researchers. For instance, the existence of libration points, their stability and the periodic orbits in the
proximity of these points under the oblateness, triaxialty
of the primaries or the effect of photogravitational force
or combination of them are studied by Sharma [3], Singh
and Ishwar [4], Sharma et al. [5, 6], Singh and Mohammed
[7], Abouelmagd and El-Shaboury [8], Abouelmagd [9],
Abouelmagd [10, 11], Abouelmagd et al. [12], Abouelmagd
and Sharaf [13], Abouelmagd et al. [14, 15], Abouelmagd et
al. [1, 16, 17, 18], Abouelmagd and Mostafa [19], Abouelmagd et al. [20], Abouelmagd and Guirao [21].
In the framework of the restricted three-body problem, for the case of rigid bodies having an axis and plane
of symmetry, the conditions for existence of collinear and
equilateral equilibrium solutions have been studied by
Vidyakin [22] and Duboshin [23].
Sharma et al. [5, 6] studied the existence and stability of libration points in the restricted three-body problem
when both the primaries are triaxial rigid bodies in the
case of stationary rotational motion (θ i = ψ i = φ i = 0).
While [24] studies the stability of infinitesimal motions
about the triangular equilibrium points in the elliptic restricted three body problem when the bigger primary is radiating and the smaller is a triaxial rigid body. They used a
technique based on Floquets Theory for determination
of characteristic exponents in the system with periodic coefficients.
The basic dynamical features of the restricted three–
body problem when the primaries are triaxial rigid bodies are studied by [25] . The equilibrium libration points
are identified and their stability is determined in the special cases when the Euler’s angles of rotational motion are
θ i = ψ i = φ i = π/2 and θ i = ψ i = π/2 , φ i = 0
, i = 1, 2 accordingly. They proved that there are three
unstable collinear equilibrium points and two triangular
such points which may be stable. Special attention has
also been paid to the study of simple symmetric periodic
orbits and 31 families consisting of such orbits have been
determined. It has been found that only one of these families consists entirely of unstable members while the re-
| 59
maining families contain stable parts indicating that other
families bifurcate from them. Finally, using the grid search
technique, a global solution in the space of initial conditions is obtained and which is comprised by simple and of
higher multiplicities symmetric periodic orbits as well as
escape and collision orbits.
In this paper, we consider the restricted three-body
problem when both primaries are triaxial rigid bodies in
two cases of stationary rotational motion θ i = 0, ψ i + φ i =
π/2 and θ i = 0, ψ i + φ i = 0 where i = 1, 2,. The necessary and sufficient conditions to find the locations of the
three collinear points are found in five cases. The linear
stability of motion in the proximity of these points is also
studied. This work is organized as follows. An overview of
the significance, some applications of the three-body problem and the aim of the present work have been discussed
in this section. A background on the restricted three-body
problem when the primaries are triaxial rigid bodies in the
general case of Euler angles of rotational motion are given
in Section 2. In Section 3 the equations of motion are found
when Euler angles of rotational motion are θ i = 0, ψ i +φ i =
π/2. In Section 4 the conditions of existence of the three libration collinear points are studied in five different cases,
while in Section 5 the linear stability of motion around the
libration collinear points are investigated. Finally, a conclusion is sketched in Section 6 and we refer to how one
can obtain the corresponding results in the case of Euler
angles of rotational motion being θ i = 0 and ψ i + φ i = 0,
i = 1, 2.
2 Background
Let (X1 , Y1 , Z1 ), (X2 , Y2 , Z2 ) and (X, Y, Z) be the coordinates of the masses m1 , m2 and m in a sidereal frame respectively. m1 and m2 are the primaries moving in a circular orbit around their center of mass and m is the mass of
the infinitesimal body that moves in the same plane of the
primaries under their gravitational field without affecting
their motion. Now we assume that the distance between
the primaries and the sum of their masses are taken equal
to unity, while the unit of time is chosen so as to make the
gravitational constant unity too. The principal axes of the
primaries are oriented to the synodic axes by Euler angles
θ i , ψ i and φ i , i = 1, 2. In addition we assume that r1 and r2
are the distances of m from m1 and m2 respectively, where
r21 = (x − µ)2 + y2 ,
r22 = (x − µ + 1)2 + y2 ,
Unauthenticated
Download Date | 6/16/17 6:48 PM
(1)
60 | F. Alzahrani, Elbaz I. Abouelmagd, Juan L.G. Guirao, and A. Hobiny
furthermore we also suppose that µ1 = m1 = 1 − µ and
µ2 = m2 = µ where µ ∈ (0, 1/2] denotes the mass ratio.
Therefore, the coordinates of the three masses m1 , m2 and
m can be written in a synodic frame as (µ, 0, 0), (µ−1, 0, 0)
and (x, y, z), correspondingly.
Since the principal axes are supposed to rotate with
the same angular velocity as that of the rigid bodies and
the bodies are moving around their center of mass without
rotation, the Euler angles remain constant throughout the
motion. Thereby the equations of motion of the infinitesimal mass m in a synodic coordinate system with dimensionless variables are governed in the form [see 5]
ẍ − 2n ẏ = Ω x ,
(2)
ÿ + 2n ẋ = Ω y ,
the elements of matrix A can be determined by the following vectors
⎛ ⎞ ⎛
⎞
a1i
− sin φ i sin ψ i + cos θ i cos φ i cos ψ i
⎜ ⎟ ⎜
⎟
⎝b1i ⎠ = ⎝− sin φ i cos ψ i − cos θ i cos φ i sin ψ i ⎠ ,
c1i
sin θ i cos φ i
⎛
⎞ ⎛
⎞
a2i
cos φ i sin ψ i + cos θ i sin φ i cos ψ i
⎜ ⎟ ⎜
⎟
⎝b2i ⎠ = ⎝cos φ i cos ψ i − cos θ i sin φ i sin ψ i ⎠ ,
c2i
sin θ i sin φ i
⎛
⎞ ⎛
⎞
a3i
− sin θ i cos ψ i
⎜ ⎟ ⎜
⎟
⎝b3i ⎠ = ⎝ sin θ i sin ψ i ⎠ .
c3i
cos θ i
See [5, 6] for details.
where Ω is the potential function which is given by [see
also 26]
[︃
]︃
2
∑︁
n2
µi
µi
2
Ω=
µr +
+
(I + I + I − 3I i ) ,
2 i i
r i 2m i r3i 1i 2i 3i
Now the potential function Ω in Eq. (3) can be rewritten in the form
Ω=
i=1
(3)
and n is the perturbed mean motion. Also, (I1i , I2i , I3i ) are
the principal moments of inertia of the triaxial rigid body
of mass m i at its center of mass with (a i , b i , c i ) as its axes,
while I i is the moment of inertia about the connected line
between the rigid body m i with the center of the infinitesimal body of mass m, and i = 1, 2 such that I ij ≠ I ji and I i
are controlled by
I i = I1i l2i + I2i m2i + I3i n2i ,
(4)
where (l i , m i , n i ) are the direction cosines of the connected line with respect to the principal axes of m i .
Now we shall adopt the notation and terminology of
[25] and follow his procedure, then we denote the unit vectors along the principal axes at m1 or m2 by (i , j , k) and
the unit vectors parallel to the synodic coordinates axes by
(e1 e2 , e3 ), and with the help of Euler angles (θ i , ψ i , φ i ),
the relation between vectors can be expressed as:
E = AQ,
where
⎛ ⎞
e1
⎜ ⎟
E = ⎝e2 ⎠
e3
⎛
,
a1i
⎜
A = ⎝a2i
a3i
b1i
b2i
b3i
⎞
c1i
⎟
c2i ⎠
c3i
(7)
,
2
∑︁
[T1i + T2i + T3i + T4i + T5i ] ,
(8)
i=1
where
]︂
µ
n2
µ i r2i + i ,
2
ri
µi
= 3 [A1i + A2i + A3i ] ,
ri
[︁
]︁2
3µ
= − 5i (A2i + A3i ) a1i (x + (−1)i µ3−i ) + a2i y ,
2r i
]︁2
3µ i [︁
= − 5 (A1i + A3i )[b1i (x + (−1)i µ3−i ) + b2i y ,
2r i
]︁2
3µ i [︁
= − 5 (A2i + A1i )[c1i (x + (−1)i µ3−i ) + c2i y ,
2r i
T1i =
T2i
T3i
T4i
T5i
[︂
(9)
and the mean motion n is governed by
n2 = 1 +
2
∑︁
[N1i + N2i ],
(10)
i=1
where
(5)
N1i = 3[A1i + A2i + A3i ],
9
N2i = − [a21i (A2i + A3i ) + b21i (A1i + A3i ) + c21i (A2i + A1i )],
2
(11)
⎛ ⎞
i
⎜ ⎟
Q = ⎝j⎠ ,
k
(6)
a2i
b2i
c2i
,
A
=
,
A
=
,
(12)
2i
3i
5R2
5R2
5R2
and R is the separation distance between the primaries,
i = 1, 2, see [25].
A1i =
Unauthenticated
Download Date | 6/16/17 6:48 PM
On the libration collinear points |
3 Equations of motion when
θ i = 0, ψ i + φ i = π/2
In the case of the Euler angels of rotational motion being
θ i = 0, ψ i + φ i = π/2, we obtain a2i = c3i = 1, b1i = −1
and the other parameters are equal to zero. The equation
of motion Eq. (2) can be rewritten in the form
ẍ − 2n ẏ = Ω x ,
ÿ + 2n ẋ = Ω y ,
(13)
where
2 [︁
∑︁
Ω=
]︁
1
2
3
4
T1i
+ T2i
+ T3i
+ T4i
,
(14)
i=1
and
1
T1i
=
(︂
µ
n2
µ r2 + i
2 i i
ri
)︂
,
µi
(A1i + A2i + A3i ),
r3i
3µ
= − 5i (A2i + A3i )y2 ,
2r i
3µ i
= − 5 (A1i + A3i )(x + (−1)i µ3−i )2 ,
2r i
2
=
T2i
3
T3i
4
T4i
(15)
2
∑︁
1
2
[N1i
+ N2i
],
(16)
i=1
where
1
N1i
= 3(A1i + A2i + A3i ),
9
2
N2i = − (A1i + A3i ),
2
simply the mean motion will take the form:
n2 = 1 +
(17)
2
3 ∑︁
(2A2i − A1i − A3i ) .
2
3. If the bigger primary is a spherical body and the
smaller is an oblate spheroid [3] (A11 = A21 = A31 =
0, A12 = A22 , A32 ≠ 0), n2 = 1 + 23 (A12 − A32 ), then
the perturbed mean motion is faster than the unperturbed motion for the oblate body with A11 > A31 .
However, If A32 > A12 (when the smaller primary is
a prolate body) the perturbed mean motion is slower
than the Keplerian motion.
4. If both primaries are oblate spheroids (A1i =
A2i , A3i ≠ 0, i = 1, 2), n2 = 1 + 23 [(A11 − A31 ) +
(A12 − A32 )] [4], then the perturbed mean motion is
faster than the unperturbed motion for oblate bodies
when A1i > A3i . However, If A3i > A1i (when the primaries are prolate bodies) the perturbed mean motion is slower than the Keplerian motion. If one of
the primaries is oblate and the other is prolate then
the perturbed mean motion will be faster or slower
than the unperturbed mean motion depending on
whether the sign of (A11 −A31 )+A12 −A32 ) is positive
or negative.
Remark 1. In the case of the rotational motion of Euler
angles being (θ i = 0, ψ i + φ i = π/2), the perturbed mean
motion will be faster or slower than the Keplerian motion
in the following cases
and the mean motion n is governed by
n2 = 1 +
61
(18)
i=1
Regarding Eq. (18), the mean motion when the rotational
motion of Euler angles are θ i = 0, ψ i + φ i = π/2 can be
analyzed for several cases:
1. If the primaries are spherical bodies (classical problem), see the book [27] (A1i = A2i = A3i = 0, i =
1, 2), thereby one obtains unperturbed mean motion n = 1.
2. If the bigger primary is an oblate spheroid and the
smaller is a spherical body [28] (A11 = A21 , A31 ≠
0 , A12 = A22 = A32 = 0), n2 = 1+ 32 (A11 − A31 ), then
the perturbed mean motion is faster than the unperturbed motion (Keplerian motion) for the oblate
body with A11 > A31 . However, If A31 > A11 (when
the bigger primary is a prolate body) the perturbed
mean motion is slower than the Keplerian motion.
1. If one of the primaries is spherical and the other is a
triaxial body, n2 = 1+ 32 (2A2i − A1i − A3i ) , (i = 1 or 2)
then the perturbed mean motion is faster or slower
than the Keplerian motion, according to whether the
sign of (2A2i − A1i − A3i ) is positive or negative.
2. If both primaries are triaxial rigid bodies, n2 = 1 +
3
2 [(2A 21 − A 11 − A 31 ) + (2A 22 − A 12 − A 32 )] then the
perturbed mean motion is faster than the Keplerian
motion when A2i > 12 (A1i + A3i ), (i = 1, 2) or (2A21 −
A11 − A31 ) + (2A22 − A12 − A32 ) > 0, otherwise it will
be slower.
4 Existence of libration collinear
points when
θ i = 0, ψ i + φ i = π/2
In general the libration points are the equilibria solutions
of the dynamical system, which describes the motion of an
infinitesimal body.
Since Ω = Ω(x, y), then
dΩ
= ẋΩ x + ẏΩ y .
dt
(19)
Unauthenticated
Download Date | 6/16/17 6:48 PM
62 | F. Alzahrani, Elbaz I. Abouelmagd, Juan L.G. Guirao, and A. Hobiny
from Eqs. (13) and (19) the Jacobi integral can be written as
2
2
ẋ + ẏ − 2Ω + C = 0.
[︃
(20)
The positions of the equilibrium points are the solutions of
the following equations
Ω x = Ω y = 0.
(21)
where
Ωx
= (x − µ)[f1 (r1 ) + q1 (x, y, r1 )]
+(x − µ + 1)[f2 (r2 ) + q2 (x, y, r2 )],
Ωy
= y[g1 (r1 ) + g2 (r2 ) + q1 (x, y, r1 ) + q2 (x, y, r2 )],
(22)
and
[︃
(︃
)︃]︃
1
3
2
f i (r i ) = µ i n −
+ 5 (2A2i + 4A1i − A3i )
,
r3i
2r i
)︃]︃
[︃
(︃
1
3
2
+ 5 (4A2i + 2A1i − A3i )
,
g i (r i ) = µ i n −
r3i
2r i
[︂
]︂
[︁
]︁2
15µ i
i
2
q i (x, y, r i ) =
A1i x + (−1) µ3−i + A2i y .
2r7i
(23)
Since the principal axes are different in triaxial rigid bodies we can assume that the triaxial rigid body of mass
m i , i = 1, 2, be nearly a sphere with radius R0i and
thereby one obtains
a i = R0i + σ1i ,
b i = R0i + σ2i ,
c i = R0i + σ3i ,
(24)
where σ1i , σ2i , σ3i ≪ 1. For investigations, see [5] and
[25].
Substituting Eqs. (24) into Eqs. (12) one obtain
A1i = λ i + δ i σ1i ,
A2i = λ i + δ i σ2i ,
A3i = λ i + δ i σ3i ,
where
(︃
)︃]︃
1 3δ i
F i (r i ) = µ i n −
+ 5 (2σ2i + 4σ1i − σ3i )
,
r3i
2r i
)︃]︃
[︃
(︃
1 3δ i
2
+ 5 (4σ2i + 2σ1i − σ3i )
,
G i (r i ) = µ i n −
r3i
2r i
[︂ [︁
]︂
]︁2
15µ i δ i
i
2
σ1i x + (−1) µ3−i + σ2i y ,
Q i (x, y, r i ) =
2r7i
(27)
and the mean motion will take the form
2
n2 = 1 +
2
3 ∑︁
δ i (2σ2i − σ1i − σ3i ) .
2
(28)
i=1
The location of the collinear points L i , i = 1, 2, 3 is determined by Ω x = Ω y = 0 and y = 0. By Eqs. (26) , (27) and
(28) this property is translated in
⎧
⎫
⎪
⎪
⎪ x + 3xδ1 (2σ21 − σ11 − σ31 )
⎪
⎪
⎪
⎪
⎪
2
⎪
⎪
⎪ 3xδ
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
+
2σ22 − σ12 − σ32 )
(
⎪
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
⎨ (1 − µ)(x − µ) µ(x − µ + 1)
⎬
−
−
3
3
f (x) =
= 0,
r
r
1
2
⎪
⎪
⎪
⎪
⎪
⎪
3δ
(1
−
µ)(x
−
µ)
1
⎪ −
⎪
⎪
(2σ21 − σ11 − σ31 ) ⎪
⎪
⎪
⎪
⎪
r51
⎪
⎪
⎪
⎪
⎪
⎪
⎪ 3δ2 µ(x − µ + 1)
⎪
⎪
⎪
⎪
⎪
−
(2σ
−
σ
−
σ
)
22
12
32
⎩
⎭
5
r2
(29)
where r1 = |x − µ| and r2 = |x − µ + 1|.
Hence we can rewrite Eq. (29) as
⎧
⎪
⎪ f1 (x) , −∞ < x < µ − 1
⎪
⎪
⎪
⎨
(30)
f (x) =
f2 (x) , µ − 1 < x < µ
⎪
⎪
⎪
⎪
⎪
⎩
f3 (x) , µ < x < ∞
where
(25)
R20i
2R0i
, δi =
.
5R2
5R2
Substituting Eqs. (25) into Eqs. (22) and Eq. (18) with the
help of Eqs. (23), we obtain
where λ i =
Ωx
= (x − µ)[F1 (r1 ) + Q1 (x, y, r1 )]
+(x − µ + 1)[F2 (r2 ) + Q2 (x, y, r2 )],
Ωy
= y[G1 (r1 ) + G2 (r2 ) + Q1 (x, y, r1 ) + Q2 (x, y, r2 )],
(26)
⎫
⎧
3xδ1
⎪
⎪
2σ
−
σ
−
σ
x
+
(
)
⎪
⎪
21
11
31
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
3xδ
2
⎪
⎪
⎪
⎪
+
2σ22 − σ12 − σ32 )
(
⎪
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
⎬
⎨
f1 (x) =
(1
−
µ)
3δ
(1
−
µ)
⎪
⎪
+
+ 1
(2σ21 − σ11 − σ31 )
⎪
⎪
⎪
⎪
(x − µ)2
(x − µ)4
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
µ
3δ
µ
⎪
⎪
2
⎭
⎩ +
+
2σ
−
σ
−
σ
( 22
12
32 )
2
4
(x − µ + 1)
(x − µ + 1)
(31)
Unauthenticated
Download Date | 6/16/17 6:48 PM
On the libration collinear points |
63
1. (σ21 ≥ σ11 ≥ σ31 ) and (σ22 ≥ σ12 ≥ σ32 ),
⎧
⎫
2.
(2σ21 − σ11 − σ31 ) ≥ 0 and (2σ22 − σ12 − σ32 ) ≥ 0,
3xδ1
⎪
⎪
x+
(2σ21 − σ11 − σ31 )
⎪
⎪
⎪
⎪
3. (2σ21 − σ11 − σ31 ) ≤ 0 and (2σ22 − σ12 − σ32 ) ≤ 0,
2
⎪
⎪
⎪
⎪
⎪
⎪
3xδ2
⎪
⎪
⎪
⎪
4.
(2σ21 − σ11 − σ31 ) ≥ 0 and (2σ22 − σ12 − σ32 ) ≤ 0,
2σ
−
σ
−
σ
+
(
)
22
12
32
⎪
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
5. (2σ21 − σ11 − σ31 ) ≤ 0 and (2σ22 − σ12 − σ32 ) ≥ 0.
⎨
⎬
f2 (x) =
(1 − µ) 3δ1 (1 − µ)
Since the parameters µ, δ i , σ1i , σ2i , σ3i as well as the
⎪
⎪
+
+
(2σ21 − σ11 − σ31 )
⎪
⎪
⎪
⎪
(x
− µ)2
(x − µ)4
⎪
⎪
⎪
⎪
quantity
1 − µ are positive, with regard to the previous five
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
cases
the
necessary and sufficient conditions can be trans⎪
⎪
3δ2 µ
µ
⎪
⎪
⎩
⎭ lated to the unequal equations (36-40) respectively.
−
−
2σ
−
σ
−
σ
(
)
22
12
32
(x − µ + 1)2 (x − µ + 1)4
(32)
σ2i ≥ σ1i ≥ σ3i ), i = 1, 2
(36)
⎫
⎧
3xδ1
⎪
⎪
x+
(2σ21 − σ21 − σ31 )
⎪
⎪
1
⎪
⎪
2
⎪
⎪
⎪
⎪
(37)
σ2i ≥ (σ1i + σ3i ), i = 1, 2
⎪
⎪
3xδ2
⎪
⎪
2
⎪
⎪
+
(2σ22 − σ12 − σ32 )
⎪
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
⎬
⎨
1
δ i |2σ2i − σ1i − σ3i | ≤ , i = 1, 2
(38)
f3 (x) =
(1
−
µ)
3δ
(1
−
µ)
1
6
⎪
⎪
−
−
(2σ21 − σ11 − σ31 )
⎪
⎪
⎪
⎪
2
4
(x − µ)
(x − µ)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1
⎪
⎪
⎪
⎪
δ2 |2σ22 − σ12 − σ32 | ≤
(39)
⎪
⎪
⎪
⎪
6
µ
3δ2 µ
⎪
⎪
⎭
⎩
−
−
2σ
−
σ
−
σ
(
)
22
12
32
(x − µ + 1)2 (x − µ + 1)4
1
(33)
δ1 |2σ21 − σ11 − σ31 | ≤
(40)
6
To investigate the existence of libration collinear points
′
′
′
and determine their locations we have to study the behav- where the derivatives f1 , f2 and f3 are given in Appendix
ior of the function f . In this context the derivative of the for every case.
Hence f ′ (x) > 0 in each of the open intervals (−∞, µ −
function f will be controlled by
⎧ ′
1), (µ−1, µ) and (µ, ∞) and it follows that f (x) is strictly inf1 (x) , −∞ < x < µ − 1
⎪
⎪
creasing in these intervals too. In addition Eq. (30) shows
⎪
⎪
⎪
⎨
that
(34)
f ′ (x) =
f2′ (x) , µ − 1 < x < µ
⎪
⎪
• Lim f (x) = −∞ and f [(µ − 1)+ ] = f (µ+ ) = −∞,
⎪
⎪
x→−∞
⎪
⎩ ′
• Lim f (x) = ∞ and f [(µ − 1)− ] = f (µ− ) = ∞.
f3 (x) , µ < x < ∞
x→∞
f1′ , f2′
f3′
where the derivatives
and
are given in the Appendix, see Eqs. (46-48).
Now Eqs. (46-48) show that the signs of the fifth
and seventh terms of the function f ′ will be effected by
the values of (2σ2i − σ1i − σ3i ) , (i = 1, 2), according
to whether both of these values are positive or negative
or with different signs, while the sum of the first three
terms is not affected and will positive all the time because
σ1i , σ2i , σ3i ≪ 1 and δ i < 1. Thereby we will investigate under which conditions f ′ (x) > 0 in the open intervals
(−∞, µ − 1), (µ − 1, µ) and (µ, ∞) and we need to find the
the necessary and sufficient conditions which makes
⎧ ′
⎪
⎪ f1 (x) > 0 , −∞ < x < µ − 1
⎪
⎪
⎪
⎨
′
f (x) =
(35)
f2′ (x) > 0 , µ − 1 < x < µ
⎪
⎪
⎪
⎪
⎪
⎩ ′
f3 (x) > 0 , µ < x < ∞
Thereby in every open interval the the f (x) changes its sign
one time from (− to + ), and we get f (µ − 2) < 0, f (0) > 0
andf (µ + 1) > 0.
Hence, we deduce that, f (µ − 2) < 0, f [(µ − 1)− ] >
0; f [(µ − 1)+ ] < 0, f (0) > 0 and f (µ+ ) < 0, f (µ + 1) > 0 and
we concludes that there are only three zeros for f (x), when
y = 0, which lie in the intervals (µ − 2, µ − 1), (µ − 1, 0)
and (µ, µ + 1). Hence, there are three collinear libration
points that lie in the intervals(µ − 2, µ − 1), (µ − 1, 0) and
(µ, µ+1), respectively. We will denote these by the symbols
L1 , L2 and L3
Now we can establish that there exists one and only
one real value for x in each of the open intervals (−∞, µ −
1), (µ − 1, µ) and (µ, ∞) such that f (x) = 0 and the function f (x) is strictly increasing in these intervals, when one
of the five conditions is achieved, otherwise we may have
more than three collinear points.
In this context we may analyze five cases, which lead
to f (x) > 0, these cases are:
′
Unauthenticated
Download Date | 6/16/17 6:48 PM
64 | F. Alzahrani, Elbaz I. Abouelmagd, Juan L.G. Guirao, and A. Hobiny
5 Stability of the libration collinear
points
To study the stability of motion around the libration
collinear points we assume that (x0 , y0 ) is one of the coordinates of these points and ξ and η are the variation
which describe the possible motion of the infinitesimal
body around one libration collinear points, where this
variation is defined as
x = x0 + ξ ,
y + y0 + η.
(41)
Substituting Eqs. (41) into Eqs. (13), in the framework of
linear stability, the variational equations will be ruled by
ξ̈ − 2n η̇ = Ω0xx ξ + Ω0xy η,
η̈ + 2n ξ̇ = Ω0xy ξ + Ω0yy η.
(42)
where the partial derivatives of the second order of Ω are
denoted by the subscripts x, y and the superscript 0 indicates that such derivative is evaluated at one of the libration collinear points. Hence the associated characteristic
equation to Eqs. (42) is
(︁
)︁
(︁
)︁2
ω4 + 4n2 − Ω0xx − Ω0yy ω2 + Ω0xx Ω0yy − Ω0xy = 0. (43)
The character of the solution of the variational dynamical system depends on the character of the solution for ω2
from the quadratic of Eq. (43). The solution is stable only
if the quadratic has two unequal negative roots for ω2 , see
[27] for more details.
From Eqs. (26), (27) and Eq. (28) as well as y = 0 at
the collinear points, the values of Ω xx , Ω xy and Ω y can be
determined by
⎫
⎧
3δ1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ 1 + 2 (2σ21 − σ11 − σ31 )
⎪
⎪
⎪
⎪
⎪
⎪
3δ
2
⎪
⎪ +
⎪
⎪
2σ
−
σ
−
σ
( 22
12
32 )
⎬
⎨
2
[︂
]︂
0
,
Ω xx =
2 3δ1
⎪
+(1 − µ) 3 + 5 (6σ21 + 48σ11 − 4σ31 ) ⎪
⎪
⎪
⎪
⎪
r
2r
⎪
⎪
1
1
[︂
]︂
⎪
⎪
⎪
⎪
⎪
⎪
2 3δ2
⎪
⎪
⎪
⎪
+µ
+
6σ
+
48σ
−
4σ
(
)
22
12
32
⎭
⎩
3
5
r2 2r2
Ω0xy = 0,
Ω0yy
⎧
3δ
⎪
⎪
1 + 1 (2σ21 − σ11 − σ31 )
⎪
⎪
2
⎪
⎪
⎪
3δ2
⎪
⎪
+
(2σ22 − σ12 − σ32 )
⎨
2
]︂
[︂
=
1 3δ1
⎪
+
4σ
−
3σ
−
σ
−(1
−
µ)
( 21
11
31 )
⎪
⎪
r31 2r51
⎪
[︂
]︂
⎪
⎪
⎪
1 3δ2
⎪
⎪
⎩ −µ 3 + 5 (4σ22 − 3σ12 − σ32 )
r2 2r2
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
.
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
(44)
For the case of (σ21 ≥ σ11 ≥ σ31 ) and (σ22 ≥ σ12 ≥
σ32 ) and additionally L1 , L2 and L3 laying in the intervals
(µ − 2, µ − 1), (µ − 1, 0) and (µ, µ + 1), respectively, we can
assume that the coordinates of L1 is (µ − 1 − ξ , 0), then r1 =
1 + ξ and r2 = ξ with 0 < ξ ≪ 1. Using Eqs. (44), we can
write Ω0xx = F(ξ ) and Ω0yy = G(ξ ) where F(ξ ) ∼
= F(0+ ) = ∞
+
0
0
∼
and G(ξ ) = G(0 ) = −∞, therefore Ω xx Ω yy < 0, and we
have Ω0xy = 0 then Ω0xx Ω0yy − (Ω0xy )2 < 0 at L1 . By the same
way we can prove that Ω0xx Ω0yy − (Ω0xy )2 < 0 for L2 and L3 .
Now we can say that the discriminant of Eq. (43) is positive under the conditions which are stated in (36-40)for
every case. Therefore, the four roots of the characteristic
Eq. (43) are real where two of them are positive and the
other two are negative. Hence the four roots are controlled
by ω1,2 = ±ω11 , ω3,4 = ±iω12 where ω11 , ω12 are reals and
i is the imaginary unit. Thereby the general solution of Eq.
(42) can be written in the form
ξ (t) =
4
∑︀
ϱ i e ωi t ,
i=1
η(t) =
4
∑︀
(45)
ρi e
ωi t
.
i=1
Remark 2. The solution in Eqs. (45) is constructed under
the first condition but we can construct this solution the
same way for the other four cases using the associated condition in every case.
Finally Eqs. (45) shows that the motion in the proximity
of libration collinear points is unbounded because ω1,2
are real and the trajectory of the infinitesimal body will include some terms, which will grow without limit. Hence
the case of instability of the libration collinear points does
not change regard to the rotational motion when Euler angles are θ i = 0, ψ i + φ i = π/2. Therefore the motion is
unstable.
Remark 3. It is worth mentioning that in the case that
the primaries are triaxial rigid bodies when the Euler angles of the rotational motion are θ i = 0, ψ i + φ i = 0,
the corresponding results can be obtained by interchanging the parameters σ11 BY σ12 and σ21 by σ22 in the cor-
Unauthenticated
Download Date | 6/16/17 6:48 PM
On the libration collinear points |
responding results given in the case of Euler angles are
θ i = 0, ψ i + φ i = π/2.
6 Conclusions
In this paper the existence of libration collinear points and
their linear stability were studied in the restricted threebody problem. This study established the setting of the primaries as being triaxial bodies in the case when the Euler
angles of rotational motion are θ i = 0, ψ i + φ i = π/2. The
necessary and sufficient conditions to determine the locations of the three collinear points are found. In addition we
show that the motion in the proximity of these points is unstable. It is worth mentioning that in the setting of the rotational motion when the primaries are triaxial rigid bodies
with θ i = 0, ψ i + φ i = 0, i = 1, 2, the corresponding results
can be obtained by interchanging the parameters σ11 and
σ21 ; σ21 and σ22 . Finally, we refer to one of the significants
of the collinear points in space missions, they are considered the optimal placement to transfer a spacecraft to an
associated stable manifold.
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎧
3δ
⎪
⎪
1 + 1 (2σ11 − σ21 − σ31 )
⎪
⎪
2
⎪
⎪
3δ2
⎪
⎪
+
(2σ12 − σ22 − σ32 )
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 2(1 − µ) 12δ1 (1 − µ)
+
+
′
f3 (x) =
(x − µ)3
(x − µ)5
⎪
⎪
⎪
(2σ11 − σ21 − σ31 )
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
12δ2 µ
2µ
⎪
⎪
+
+
⎪
⎪
3
⎪
(x
−
µ
+
1)
(x
− µ + 1)5
⎪
⎩
(2σ12 − σ22 − σ32 )
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
(47)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
(48)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
Case 3:
When (2σ21 − σ11 − σ31 ) ≤ 0 and (2σ22 − σ12 − σ32 ) ≤ 0
Appendix
Case 1 and 2:
When (σ21 ≥ σ11 ≥ σ31 ) and (σ22 ≥ σ12 ≥ σ32 ) or
(2σ21 − σ11 − σ31 ) ≥ 0 and (2σ22 − σ12 − σ32 ) ≥ 0,
⎧
3δ
⎪
⎪
1 + 1 (2σ21 − σ21 − σ31 )
⎪
⎪
2
⎪
⎪
3δ2
⎪
⎪
+
(2σ22 − σ22 − σ32 )
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 2(1 − µ) 12δ1 (1 − µ)
−
−
′
f1 (x) =
(x − µ)3
(x − µ)5
⎪
⎪
⎪
(2σ21 − σ21 − σ31 )
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2µ
12δ2 µ
⎪
⎪ −
−
⎪
⎪
⎪
(x − µ + 1)3 (x − µ + 1)5
⎪
⎩
(2σ22 − σ12 − σ32 )
⎧
3δ
⎪
⎪
1 + 1 (2σ11 − σ21 − σ31 )
⎪
⎪
2
⎪
⎪
3δ2
⎪
⎪
+
(2σ12 − σ22 − σ32 )
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 2(1 − µ) 12δ1 (1 − µ)
−
−
′
f2 (x) =
(x − µ)3
(x − µ)5
⎪
⎪
⎪
2σ
−
σ
−
σ
(
11
21
31 )
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2µ
12δ2 µ
⎪
⎪
+
+
⎪
⎪
3
⎪
(x
−
µ
+
1)
(x
− µ + 1)5
⎪
⎩
(2σ12 − σ22 − σ32 )
65
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
(46)
⎫
⎧
)︀
3δ1 ⃒⃒
⎪
⎪
⎪
⎪
1
−
(2σ
−
σ
−
σ
|
21
21
31
⎪
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
⃒
)︀
3δ2 ⃒
⎪
⎪
⎪
⎪
⎪
⎪
−
(2σ
−
σ
−
σ
|
22
22
32
⎪
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎨
⃒
)︀
′
12δ
(1
−
µ)
2(1
−
µ)
1
⃒(2σ21 − σ21 − σ31 |
f1 (x) =
+
−
⎪
⎪
(x − µ)3
(x − µ)5
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
12δ
µ
2µ
2
⎪
⎪
⎪
⎪
−
+
⎪
⎪
3
5
⎪
⎪
(x
−
µ
+
1)
(x
−
µ
+
1)
⎪
⎪
⃒
⎪
⎪
)︀
⎭
⎩ ⃒
(2σ22 − σ12 − σ32 |
(49)
⎧
⎫
)︀
3δ1 ⃒⃒
⎪
⎪
⎪
⎪
1
−
(2σ
−
σ
−
σ
|
11
21
31
⎪
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
⃒
)︀
3δ2 ⃒
⎪
⎪
⎪
⎪
⎪
⎪
−
(2σ
−
σ
−
σ
|
12
22
32
⎪
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎨
⃒
)︀
′
2(1
−
µ)
12δ
(1
−
µ)
1
⃒(2σ11 − σ21 − σ31 |
f2 (x) =
−
+
⎪
⎪
(x − µ)3
(x − µ)5
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2µ
12δ
µ
2
⎪
⎪
⎪
⎪
+
−
⎪
⎪
3
5
⎪
⎪
(x
−
µ
+
1)
(x
−
µ
+
1)
⎪
⎪
⃒
⎪
⎪
)︀
⎭
⎩ ⃒
(2σ12 − σ22 − σ32 |
(50)
Unauthenticated
Download Date | 6/16/17 6:48 PM
66 | F. Alzahrani, Elbaz I. Abouelmagd, Juan L.G. Guirao, and A. Hobiny
When (2σ21 − σ11 − σ31 ) ≤ 0 and (2σ22 − σ12 − σ32 ) ≥ 0
⎧
)︀
3δ ⃒
⎪
⎪
1 − 1 ⃒(2σ11 − σ21 − σ31 |
⎪
⎪
2
⎪
⎪
)︀
3δ ⃒
⎪
⎪
− 2 ⃒(2σ12 − σ22 − σ32 |
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 2(1 − µ) 12δ1 (1 − µ)
−
+
′
3
f3 (x) =
(x −)︀µ)5
⃒ (x − µ)
⎪
⎪
⃒
⎪
(2σ
−
σ
−
σ31 |
11
21
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
12δ2 µ
2µ
⎪
⎪
−
+
⎪
⎪
3
5
⎪
(x
−
µ
+
1)
(x
⎪ ⃒
)︀− µ + 1)
⎩
⃒(2σ12 − σ22 − σ32 |
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎧
)︀
3δ ⃒
⎪
⎪
1 − 1 ⃒(2σ21 − σ21 − σ31 |
⎪
⎪
2
⎪
⎪
3δ
⎪
⎪
+ 2 (2σ22 − σ22 − σ32 )
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 2(1 − µ) 12δ1 (1 − µ)
+
−
′
3
f1 (x) =
(x −)︀µ)5
⃒ (x − µ)
⎪
⎪
⃒
⎪
(2σ
−
σ
−
σ31 |
21
21
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
12δ2 µ
2µ
⎪
⎪
−
−
⎪
⎪
3
⎪
(x
−
µ
+
1)
(x
− µ + 1)5
⎪
⎩
(2σ22 − σ12 − σ32 )
(51)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
Case 4:
When (2σ21 − σ11 − σ31 ) ≥ 0 and (2σ22 − σ12 − σ32 ) ≤ 0
Case 5:
⎧
3δ
⎪
⎪
1 + 1 (2σ21 − σ21 − σ31 )
⎪
⎪
2
⎪
⎪
)︀
3δ2 ⃒⃒
⎪
⎪
(2σ22 − σ22 − σ32 |
−
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 2(1 − µ) 12δ1 (1 − µ)
−
−
′
f1 (x) =
(x − µ)3
(x − µ)5
⎪
⎪
⎪
2σ
−
σ
−
σ
( 21
21
31 )
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2µ
12δ2 µ
⎪
⎪
−
+
⎪
⎪
3
5
⎪
(x
−
µ
+
1)
(x
⎪
)︀− µ + 1)
⎩ ⃒⃒
(2σ22 − σ12 − σ32 |
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎧
3δ
⎪
⎪
1 + 1 (2σ11 − σ21 − σ31 )
⎪
⎪
2
⎪
⎪
)︀
3δ2 ⃒⃒
⎪
⎪
−
(2σ12 − σ22 − σ32 |
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 2(1 − µ) 12δ1 (1 − µ)
−
−
f2′ (x) =
(x − µ)3
(x − µ)5
⎪
⎪
⎪
(2σ11 − σ21 − σ31 )
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2µ
12δ2 µ
⎪
⎪
−
+
⎪
⎪
⎪ ⃒ (x − µ + 1)3 (x )︀− µ + 1)5
⎪
⎩ ⃒
(2σ12 − σ22 − σ32 |
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎧
3δ
⎪
⎪
1 + 1 (2σ11 − σ21 − σ31 )
⎪
⎪
2
⎪
⎪
)︀
3δ2 ⃒⃒
⎪
⎪
−
(2σ12 − σ22 − σ32 |
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 2(1 − µ) 12δ1 (1 − µ)
+
+
f3′ (x) =
(x − µ)3
(x − µ)5
⎪
⎪
⎪
(2σ11 − σ21 − σ31 )
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2µ
12δ2 µ
⎪
⎪
+
−
⎪
⎪
⎪ ⃒ (x − µ + 1)3 (x )︀− µ + 1)5
⎪
⎩ ⃒
(2σ12 − σ22 − σ32 |
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎧
)︀
3δ ⃒
⎪
⎪
1 − 1 ⃒(2σ11 − σ21 − σ31 |
⎪
⎪
2
⎪
⎪
3δ
⎪
⎪
+ 2 (2σ12 − σ22 − σ32 )
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 2(1 − µ) 12δ1 (1 − µ)
+
−
′
f2 (x) =
(x − µ)3
(x −)︀µ)5
⃒
⎪
⎪
⃒
⎪
(2σ11 − σ21 − σ31 |
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2µ
12δ2 µ
⎪
⎪
+
+
⎪
⎪
3
⎪
(x
−
µ
+
1)
(x
− µ + 1)5
⎪
⎩
(2σ12 − σ22 − σ32 )
(52)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
(54)
(55)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
(56)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎧
)︀
3δ ⃒
⎪
⎪
1 − 1 ⃒(2σ11 − σ21 − σ31 |
⎪
⎪
2
⎪
⎪
3δ
⎪
⎪
+ 2 (2σ12 − σ22 − σ32 )
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 2(1 − µ) 12δ1 (1 − µ)
−
+
f3′ (x) =
(x − µ)3
(x −)︀µ)5
⃒
⎪
⎪
⃒
⎪
(2σ11 − σ21 − σ31 |
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2µ
12δ2 µ
⎪
⎪
+
+
⎪
⎪
⎪ (x − µ + 1)3 (x − µ + 1)5
⎪
⎩
(2σ12 − σ22 − σ32 )
(53)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
(57)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
Acknowledgement: This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (36 - 130 - 36 RG). The authors,
therefore, acknowledge with thanks DSR technical and financial support. Furthermore, this work was partially supported by MICINN/FEDER: grant number MTM2011-22587;
MINECO: grant number MTM2014-51891-P, and Fundación
Séneca de la Región de Murcia: grant number 19219/PI/14.
References
[1]
Abouelmagd E.I., Alhothuali M.S., Guirao J.L.G., Malaikah H.M.,
The effect of zonal harmonic coeflcients in the framework of the
Unauthenticated
Download Date | 6/16/17 6:48 PM
On the libration collinear points |
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
restricted three-body problem, Adv. Space Res., 2015, 55, 16601672.
Koon W.S., Lo M.W. , Marsden J.E., Ross S.D., Constructing a
Low Energy Transfer between Jovian Moons-Proceedings of the
International Conference on Celestial Mechanics (15-19 December 1999, Northwestern University, Evanston, Illinois), American
Mathematical Soc., 2002, 129-143.
Sharma R.K., The linear stability of libration points of the photogravitational restricted three-body problem when the smaller
primary is an oblate spheroid, Ap&SS, 1987, 135, 271-281.
Singh J., Ishwar B., Stability of triangular points in the photogravitational restricted three body problem, ChA&A, 1999, 27,
415-424.
Sharma R.K., Taqvi Z.A., Bhatnagar, K.B., Existence of libration
Points in the restricted three body problem when both primaries
are triaxial rigid bodies, Indian J. Pure Appl. Math., 2001, 32(1),
125-141.
Sharma R.K., Taqvi Z.A., Bhatnagar K.B., Existence of libration
Points in the restricted three body problem when both primaries
are triaxial rigid bodies and source of radition, Indian J. Pure
Appl. Math., 2001, 32(7), 981-994.
Singh J., Mohammed H.L., Robe’s Circular Restricted ThreeBody Problem Under Oblate and Triaxial Primaries, Earth Moon
Planets, 2012, 109, 1-11.
Abouelmagd E.I., El-Shaboury S.M., Periodic orbits under combined effects of oblateness and radiation in the restricted problem of three bodies, Ap&SS, 2012, 341, 331-341.
Abouelmagd E.I., Existence and stability of triangular points in
the restricted three-body problem with numerical applications,
Ap&SS, 2012, 342, 45-53.
Abouelmagd E.I., Stability of the triangular points under combined effects of radiation and oblateness in the restricted threebody problem, Earth Moon Planets, 2013, 110, 143-155.
Abouelmagd E.I., The effect of photogravitational force and
oblateness in the perturbed restricted three-body problem,
Ap&SS, 2013, 346, 51-69.
Abouelmagd E.I., Asiri H.M., Sharaf M. A., The effect of oblateness in the perturbed restricted three–body problem, Meccanica, 2013, 48, 2479-2490.
Abouelmagd E.I., Sharaf M.A., The motion around the libration
points in the restricted three-body problem with the effect of
radiation and oblateness, Ap&SS, 2013, 344, 321-332.
Abouelmagd E.I., Awad M.E., Elzayat E.M.A., Abbas I.A., Reduction the secular solution to periodic solution in the generalized
restricted three-body problem, Ap&SS, 2014, 350, 495-505.
67
[15] Abouelmagd E.I., Guirao, J.L.G., Mostafa A., Numerical integration of the restricted three-body problem with Lie series,
Ap&SS, 2014, 354, 369-378.
[16] Abouelmagd E.I., Alhothuali M.S., Guirao J.L.G., Malaikah H.M.,
On the periodic structure in the planar photogravitational Hill
problem, Applied Mathematics & Information Science, 2015,
9(5), 2409-2416.
[17] Abouelmagd E.I., Alhothuali M.S., Guirao J.L.G., Malaikah H.M.,
Periodic and secular solutions in the restricted three body problem under the effect of zonal harmonic parameters, Applied
Mathematics & Information Science, 2015, 9(4), 1-11.
[18] Abouelmagd E.I., Mostafa A., Guirao J.L.G., A first order automated Lie transform, International Journal of Bifurcation and
Chaos, 2015, 25 (14), 1540026.
[19] Abouelmagd E.I., Mostafa A., Out of plane equilibrium points locations and the forbidden movement regions in the restricted
three-body problem with variable mass, Ap&SS, 2015, 357, 58.
[20] Abouelmagd E.I., Alzahrani F., Guirao J.L.G., Hobiny A., Periodic
orbits around the collinear libration points, J. Nonlinear Sci.
Appl. (JNSA), 2016, 9(4), 1716-1727.
[21] Abouelmagd E.I., Guirao J.L.G., On the perturbed restricted
three-body problem, Applied Mathematics and Nonlinear Sciences, 2016, 1(1), 123-144.
[22] Vidyakin V.V., The plane restricted circular problem of three
spheroids, Soviet Astronomy, 1975, 18(5), 641-645.
[23] Duboshin G.N., Sur les mouvements rguliers dans le
problme des deux corps solides (On regular solutions in the
problems of two rigid bodies), Celes. Mech. Dyn. Astr., 1982,
27(3), 267-284.
[24] Narayan A., Usha T., Stability of triangular equilibrium points in
the elliptic restricted problem of three bodies with radiating and
triaxial primaries, Ap&SS, 2014, 351(1), 135-142.
[25] Elshaboury S.M., Abouelmagd E.I., Kalantonis V.S., Perdios
E.A., The planar restricted three-body problem when both primaries are triaxial rigid bodies: Equilibrium points and periodic
orbits, Ap&SS, 2016, 361(9), 315.
[26] McCusky S.W., Introduction to Celestial Mechanics, Addision
Wesley, 1963.
[27] Szebehely V., Theory of orbits: the restricted three body problem, Academic Press, 1967.
[28] Bhatnagar K.B., Hallan P.P., Effect of perturbed potentials on
the stability of libration points in the restricted problem, Celes.
Mech. Dyn. Astr., 1979, 20, 95-103.
Unauthenticated
Download Date | 6/16/17 6:48 PM